The defender will sign a contract until 30 June 2026 and his buy-out clause will be set at 500 million euros (£430m).
Christensen becomes the second Chelsea centre-back to depart on a free and head to Spain following Antonio Rudiger's move to Real Madrid.
The news comes shortly after Barcelona announced the signing of Franck Kessie on a free transfer.
The Ivory Coast midfielder's deal with AC Milan expired in June after he helped them to win their first Serie A title in 11 years.
